Wet Bulb Globe Temperature: Indicating Extreme Heat Risk on a Global Grid

Abstract The Wet Bulb Globe Temperature (WBGT) is an international standard heat index used by the health, industrial, sports, and climate sectors to assess thermal comfort during heat extremes. Observations of its components, the globe and the wet bulb temperature (WBT), are however sparse. Therefore WBGT is difficult to derive, making it common to rely on approximations, such as the ones developed by Liljegren et al. (2008, https://doi.org/10.1080/15459620802310770, WBGTLiljegren) and by the American College of Sports Medicine (WBGTACSM87). In this study, a global data set is created by implementing an updated WBGT method using ECMWF ERA5 gridded meteorological variables and is evaluated against existing WBGT methods. The new method, WBGTBrimicombe, uses globe temperature calculated using mean radiant temperature and is found to be accurate in comparison to WBGTLiljegren across three heatwave case studies. In addition, it is found that WBGTACSM87 is not an adequate approximation of WBGT. Our new method is a candidate for a global forecasting early warning system.

Whereas 2 m air temperature is easily measurable, observations of globe thermometer and wet bulb thermometer temperatures are often sparse (Budd, 2008;D'Ambrosio Alfano et al., 2014). Consequently, it has been historically challenging to calculate WBGT from Equation 1 and it is instead common to rely on approximations. These include the approximation from the American College of Sports Medicine (termed WBGTACSM87 ), which is a linear model of the WBGT (American college of sports medicine, 1987), and the approximation by Liljegren and colleagues (termed WBGTLiljegren ), which is a more complex approximation based on the fundamentals of heat transfer (Liljegren et al., 2008).
In this study, we compare a new approach to approximate WBGT (termed WBGTBrimicombe ) with WBGTACSM87 and WBGTLiljegren . Our approach is novel in calculating WBGT from gridded data using the variable of mean radiant temperature and is designed for operational forecasting systems. Comparisons are performed globally by using the ERA5 hourly global gridded reanalysis from the European Centre for Medium-Range Weather Forecasts (ECMWF), Observation data from the World Radiation Monitoring Center-Baseline Surface Radiation Network (Driemel et al., 2018) and are here discussed within the context of three heatwave case studies (India and Pakistan in July 2003, the Western Sahel in March 2013 and Australia in December 2019).

Brimicombe WBGT Approximation (WBGT Brimicombe )
This new approach to approximate WBGT has been developed for numerical weather prediction post-processing as it takes an optimized approach to the calculation of WBGT by removing the need for iterative loops. We calculate globe temperature using an adapted version of the original Bedford and Warner equation, making use of mean radiant temperature, a measurement of incidence of radiation on a body which is appropriate for indoor or outdoor use depending on given inputs (Bedford & Warner, 1934;De Dear, 1987;Guo et al., 2018;Thorsson et al., 2007;Vanos et al., 2021).
Here Equation 2 is used to solve for globe temperature as the subject because the ERA5 reanalysis data contains the variables of 2 m air temperature (T a ), 10-m wind speed (v a ), and mean radiant temperature (T MRT ). All temperatures are in Kelvin; 10-m wind speed was found to be within ±1°C of an approximated 2 m wind speed which used the method found in Spangler et al. (2022) and therefore is used (not shown). The code to compute this is available as part of thermofeel: https://doi.org/10.21957/mp6v-fd16 (Brimicombe et al., 2021;Brimicombe, Di Napole et al., 2022) In Equation 2, cg is the mean convection coefficient and is calculated using Equation 3. This is an additional correction from the original method and reduces the impact weighting of high wind speeds on the outputted globe temperature (De Dear, 1987;Guo et al., 2018).
To calculate the wet bulb temperature (WBT), a theoretical method by Stull (2011) is used and is shown in Equation 4, where T a is 2 m air temperature in °C and RH is relative humidity in percent. This method is valid between −20°C and 50°C and between 5% and 99% humidity, which are the ranges the method is optimized for and with which it has been used in previous studies (Freychet et al., 2020;Heo et al., 2019;Raymond et al., 2017). In addition, this method provided a test case, an expected value for a given set of inputs, which allowed validation of the calculated value (Stull, 2011

Liljegren WBGT Approximation (WBGT Liljegren )
WBGT Liljegren can be considered a existing "gold standard" benchmark WBGT value as it is widely considered the most accurate WBGT approximation available (Kjellstrom et al., 2009;Kong & Huber, 2021;Liljegren et al., 2008). To obtain WBGT Liljegren , WBT is calculated as per Equation 5 and globe temperature is calculated as per Equation 6 which are then used in Equation 1. Specifically, WBT is calculated as where Nu is the Nusselt number, Sh is the Sherwood Number, Pr is the Prandtl number, and Sc is the Schmidt is the change in saturation water vapor transfer between the hygrometer wick and its surroundings.
∆ net ℎ is the net radiative heat flux divided by the convective heat transfer coefficient. Full details can be seen in (Liljegren et al., 2008). Globe temperature is calculated as where the ℎ g ( g − a ) and 2 g ( 1 − g ) terms denote the energy gain from diffuse downwards and direct downward solar radiation respectively. dsrp is the projected area and sf c is the reflected solar radiation and ssrd is downward solar radiation. Full details can be seen in Liljegren et al. (2008) . Here data for WBGT Liljegren is provided by Kong and Huber (2021), instead of calculation using the HEAT-SHIELD methodology, because the method presented by Kong and Huber appears to be more robust and closer to the original methodology (Casanueva, 2017).

Methodological Difference Between WBGT Brimicombe and WBGT Liljegren
Several methodological differences between our new WBGT approximation and the existing "gold standard" WBGT approximation are present Liljegren et al., 2008).
One key variable that is necessary in the calculation of T g (both in Equations 2 and 6) is the cosine of the solar zenith angle. In previous studies it is found that radiation in the Liljegren T g methodology has inaccuracies at sunrise and sunset due to the method used to calculate this variable (Kong & Huber, 2022;Lemke & Kjellstrom, 2012). These inaccuracies are known to become greater in a numerical weather prediction service time step (a period of several hours) Hogan & Hirahara, 2016). For the Brimicombe T g methodology this does not occur because a specially designed cosine of the solar zenith angle is implemented .
Another difference is in the number of radiation input variables used to calculate T g in the Liljegren methodology. In this only 2 radiation components are used (Liljegren et al., 2008) in comparison to the 5 that calculate T MRT (please refer to: Di Napoli et al., 2020) which goes on to calculate the Brimicombe T g . Equation 2, which expresses mean radiant temperature T MRT as a function of T g and T a , is comparable to the heat balance expressed in Bedford and Warner (1934), therefore relating mean radiant temperature to the temperature of surrounding surfaces. Similarly, Equation 2 can also be rearranged in order of T g 4 , where many comparable terms to Equation 6 are identifiable.
In the Liljegren T w methodology a psychrometric WBT is calculated using fundamentals of mass transfer. In addition a key input of saturation water vapor pressure is calculated differently over ice and water (and the land 4 of 14 surface) as in Hardy (1998). In comparison, in the Brimicombe T w methodology an empirical theoretical WBT is calculated (Stull, 2011). As previously mentioned, this computationally removes the need for iterative loops, which are onerous to run for a gridded data set. In addition, the saturation water vapor pressure method is only for over water (and the land surface) in contrast to being over either water or ice, given that WBGT is a human heat stress index. How these methodological differences introduce errors will be explored within this study.

WBGT Approximation Comparisons
To compare the WBGT approximations, this study uses variables available as part of the ERA5 and ERA5-HEAT gridded reanalysis data sets produced by ECMWF on a 0.25° × 0.25° grid at an hourly time step Hersbach et al., 2020). ERA5 was chosen for this study as a state-of-the-art gridded reanalysis data set; it is an ideal data set to test out a new gridded based methodology and has the added benefit of outputting the mean radiant temperature variable (the incidence of radiation on the body). ERA5 and ERA5-HEAT variables of 2 m air temperature, 2 m dew point temperature, 10 m wind speed and mean radiant temperature are used in the relevant equations to calculate WBGTBrimicombe and WBGTACSM87 . WBGTLiljegren is also calculated using ERA5 reanalysis data. There are known limitations of ERA5; these include inaccuracies at higher elevations (Brunamonti et al., 2019;Senyunzi et al., 2020).
The approximations are calculated and compared for three past heatwaves on dates where heat stress is known to have occurred. One affected India and Pakistan in July 2003, another the Western Sahel in March 2013 and another Australia in December 2019 (CRED, 2020). In addition, this study also considers the full global gridded data sets of WBGT values including those below a heat stress threshold.
The comparison between the three approximations uses the observed WBGT thresholds set out by the ISO (Jacklitsch et al., 2016; Table 1). According to ISO, which considers heat stress by reference to recommended lifting and hard labor workloads, 33°C is known as a critical health threshold for WBGT (Heo et al., 2019). WBGTACSM87 and WBGTBrimicombe are compared against the existing gold standard approximation of WBGTLiljegren in two ways. The first is through evaluation of the spatial anomaly in WBGT values. WBGTLiljegren is subtracted from the corresponding WBGTACSM87 or WBGTBrimicombe values. Second, is a correlation between WBGTLiljegren and the other WBGT approximations is assessed together with the mean absolute error (MAE). In addition the sensitivity of the outputted WBGTBrimicombe and WBGTLiljegren approximations to key input variables is assessed.

Gridded Outputs of WBGT for the Three Heatwaves
In   (Figure 1, right column). WBGTACSM87 also performs better for this heatwave than for the July 2003 and March 2013 heatwaves and heat stress values are close to those of WBGTLiljegren . However, WBGTACSM87 again does not capture the same shape of the areas under heat stress. These similarities and differences can also be seen clearly at the global scale for each heatwave (Figure 2). WBGTLiljegren and WBGTBrimicombe values are similar worldwide in each of the 3 months considered (Figure 2), particularly focusing on parts of North Africa, southern Asia and Australia. It is however noteworthy that WBGTBrimicombe does not always capture the highest heat stress category indicated by WBGTLiljegren in South

WBGT Approximations Anomalies
Overall, the anomalies between WBGTLiljegren and WBGTBrimicombe are small, with negative anomalies indicating where WBGTLiljegren has higher values than WBGTBrimicombe , the term anomaly is used to denote deviations of WBGT approximations in comparison to the current gold standard WBGTLiljegren . In July 2003, WBGT Liljegren has higher values than WBGT Brimicombe across most of the land surface (Figure 3, left column). In addition, anomalies can be seen to be no more or less than ±2°C, except in Greenland which has anomalies of up to −4°C. In comparison, March 2013 has a similar pattern where anomalies can be seen to not be more or less than ±2°C between WBGT Liljegren and WBGT Brimicombe (Figure 3, middle column). However, for March 2013, more of the northern hemisphere has anomalies of −4°C, for example, in Canada and Siberia colder regions. Fewer regions experience anomalies of −4°C for December 2019, with this only present in the Himalaya into Tibet and the Canadian Rockies regions of higher elevation (Figure 3, right column). Overall across each of the three case studies, most of the land surface has anomalies of only ±2°C between WBGT Liljegren and WBGT Brimicombe .
For WBGT Liljegren in comparison to WBGT ACSM87 averaging across all years for the southern hemisphere, WBGT Liljegren have values higher than WBGT ACSM87 by 4°C. In contrast, for the March 2013 and December 2019 heatwaves, the northern hemisphere has anomalies of +4°C. Anomalies are less in the Sahara desert and Australia. For the July 2003 heatwave, anomalies match with those seen in the Southern hemisphere of −4°C.

WBGT Approximations Correlations
Across both WBGT ACSM87 and WBGT Brimicombe there is a strong linear correlation to WBGT Liljegren (Figure 4). WBGT Brimicombe has smaller MAE values across all three case studies than WBGT ACSM87 , with the smallest value  Table 1. Sea area has been masked. 7 of 14 being 0.76°C, being on average smaller for values about the heat stress threshold. WBGT ACSM87 MAE values are large and range between 3.39°C and 5.51°C across the three case studies but are significantly smaller above the heat stress threshold ranging from 2.87°C to 3.11°C. WBGT Brimicombe heat stress values (above 23°C, category 1 onwards) have a stronger linear relationship than across the whole distribution. In comparison, WBGT ACSM87 has a bigger spread in the cluster of points than WBGT Brimicombe over the whole distribution.

WBGT Approximation Differences
It has already been demonstrated that WBGT ACSM87 differs significantly from the other WBGT approximations presented. Therefore, here the sensitivity of only WBGT Liljegren and WBGT Brimicombe to key input variables is shown in more depth. Figure 5 demonstrates that broadly for both WBGT Liljegren and WBGT Brimicombe high solar radiation, temperature, humidity with low wind speeds lead to the highest WBGT values. In Figure 5   with the outputted WBGT Liljegren . This confirms the trend is due to the WBGT Liljegren Saturation Water Vapor pressure method (Figure 4) . As suggested in Section 2.4 this discrepancy comes from the difference in the Tw methodology, specifically from how saturation vapor pressure is calculated.
The sensitivity of WBGT Liljegren and WBGT Brimicombe is highly similar for solar radiation and wind speed and can be suggested to provide further evidence that Equations 2 and 6 are comparable. This is despite the potential discrepancies that were suggested in Section 2.3. This should be explored further to inform more about the inter-dependencies of the different types of radiation. Further we find that WBGT does not have a dynamical response to wind similar to previous findings and this can be suggested to be a limitation of the heat stress index (Foster et al., 2022).

WBGT Brimicombe Observations Comparisons
WBGT Brimicombe for reanalysis data performs robustly in comparison to WBGT Liljegren it also performs accurately compared to WBGT Brimicombe observed ( Figure 6). WBGT Brimicombe has R 2 values between its ERA5 values and observation values of between 0.56 and 1 (Figures 6b-6e). It performs better for the TAT station (Tateno) situated in Japan, where WBGT values are higher than the NYA (Nya Långenäs) station situated in the Artic circle in Svalbard overall. MAE values range from 0.97°C to 5.06°C (Figures 6b-6e). The R 2 and error values are comparable with those seen between observed MRT and ERA5 MRT in Di Napoli et al. (2020).
In addition, when evaluating the Stull method in comparison to the Davies-Jones method to calculate WBT for March 2013 in the observed data small differences are observed ( Table 2). The biggest MAE value is 1.43°C in WBT for NYA decreasing to 1 C in WBGT. The least significant R 2 value is 0.61 for WBT for TAT. It therefore can be suggested that using Stull in comparison to Davies-Jones makes no substantial difference in the resulting WBGT.

Why Another WBGT Approximation?
We demonstrate that WBGTBrimicombe is a useful approximation of WBGT. As discussed in Section 2.4 and supported by the results in Section 3.4, WBGTBrimicombe is a beneficial method to use in the place of WBGTLiljegren for gridded data sets and numerical weather prediction services. Comparisons between WBGTBrimicombe and WBGTLiljegren show only small differences (a difference of 1 heat stress category and a MAE of between 0.76°C and 1.13°C) across the case studies considered.
WBGTBrimicombe reanalysis has at most an MAE value of 5°C in comparison to it being observed ( Figure 6).
WBGTBrimicombe performs with the least accuracy in cold climates such as Greenland and at higher altitudes such as the Tibetan Plateau, regions that are not highly populated and are cold which is outside the scope of a heat stress index ( Figure 3). As such, it has been shown with confidence that WBGTBrimicombe can be considered an accurate approximation of WBGT (Figures 1-6). There are many approaches to calculating the WBGT and these derive from the fact that measurements from globe and wet bulb thermometers are not widely available (Dally et al., 2018;Lemke & Kjellstrom, 2012;Lima et al., 2021;Orlov et al., 2020;Yengoh & Ardö, 2020). Unless measurements come from these instruments and provide all the input parameters required in Equation 1, all approaches to calculate the WBGT are approximations, which are wide ranging in accuracy, a wide scale observation study, in terms of both weather and physiological observations would therefore be beneficial. This research, however, clearly demonstrates that the approximation by the American college of sports medicine, WBGTACSM87 , is not an accurate indication of WBGT and recommends that it is not used for a like-for-like approximation. This finding is in agreement with current literature on the topic (Chen et al., 2019;Grundstein & Cooper, 2018;Kong & Huber, 2021;Lemke & Kjellstrom, 2012;Lima et al., 2021;Orlov et al., 2020;Yengoh & Ardö, 2020).
Previous research has suggested that the approximation by Davies-Jones (2008) is a more accurate approximation of natural WBT than the approximation by Stull (Buzan et al., 2015). However, the results presented here demonstrate the accuracy of the WBGTBrimicombe results and the similar sensitivity of this approximation using Stull (2011)

in comparison to
WBGTLiljegren and observed calculations of WBGTBrimicombe . Further, for observation data it has been shown by this study that there is no more than a 1°C MAE between a WBGT output using Davies-Jones (2008) in comparison to Stull (2011). In addition, and of particularly practical relevance, the approximation by Stull (2011) is not iterative and therefore easier to use and more readily scalable than the Davies-Jones approximation.
WBGTBrimicombe was developed for gridded data sets from numerical weather 10 of 14 prediction data sets and is as accurate as WBGTLiljegren whilst removing the need for complicated iterative convergence methods that can practically take a long time to run and are not readily designed for gridded data. Given all of this evidence it is unnecessary to assess Davies-Jones further by this study.

How Useful Are Set Thresholds for WBGT?
WBGT ACSM87 was found to be significantly lower than WBGTLiljegren for heat stress categories and overall is not an accurate indication of WBGT heat stress risk (as per Kong and Huber (2021)). This could be of particular disadvantage to the health sector where thresholds are often used to identify life-threatening conditions or to recommend heat-suitable workloads (Budd, 2008;Chen et al., 2019;Jendritzky et al., 2012;Zare et al., 2019).
In this study, it is demonstrated that WBGTBrimicombe can use the same thresholds to indicate heat stress as WBGTLiljegren with these being meaningful values for hazard preparedness. The deliberate decision is taken to use heat stress categories for WBGT as set out by Jacklitsch et al. (2016), where the highest value of 33°C has been shown to be a critical level for heat stress illnesses and to correlate with an increase in hospital admissions and mortality (Cheng et al., 2019). Many studies assessing heat stress and extreme heat are now making use of percentiles compared to a climate (Guigma et al., 2020;Heo et al., 2019) or a standard deviation compared to average conditions (Harrington & Otto, 2020). Whilst we acknowledge that heat indexes and their studies, as the present one, often still do not take into account acclimatization and that 26°C will not be experienced the same by someone in the UK in comparison to Australia (Buzan & Huber, 2020;Nazarian & Lee, 2021), we see the categorical approach as fundamental to heat hazard preparedness. We support more research into acclimatization and how to best model this with heat stress thresholds and health outcomes in mind.

The Use of WBGT in Weather Forecasting
The WBGT is widely used across sectors. Our approach to the WBGT has been validated in its component parts, namely in the globe thermometer temperature and the wet bulb thermometer temperature (De Dear, 1987;Guo et al., 2018;Stull, 2011). It has been demonstrated for the first time (Section 3.4) that the T g method of

WBGTBrimicombe and
WBGTLiljegren are comparable. Going forward this could be used to inform more about radiation. In addition, it is designed for easy integration into operational weather prediction outputs and for use with gridded data sets, with a view to forecast heat stress and heatwaves on a global scale .
The ISO status of the WBGT makes it stand out as a heat index that is worth forecasting across multiple sectors (Heo et al., 2019). It is important to forecast WBGT to inform decisions about heat stress warnings and adaptations. There are many benefits when forecasts are made openly accessible and many factors to consider (Budd, 2008;Buzan et al., 2015;Lemke & Kjellstrom, 2012). These include: the accuracy of a WBGT approximation in comparison to the ISO observed values used in Equation 1; the robustness of thresholds in indicating heat hazards and heat stress risk levels; the appropriateness of WBGT for different climates and acclimatization levels (Ahn et al., 2022;Budd, 2008;D'Ambrosio Alfano et al., 2014). These factors also hold true for other heat indices and should be carefully considered (Ahn et al., 2022;Zare et al., 2019).

Conclusion
WBGT Brimicombe has been demonstrated to be an accurate approximation of WBGT. WBGT Brimicombe is within 1 heat stress category of WBGT Liljegren across the land surface and in general has anomalies of no more than ±2°C for the 3 heatwave case studies here chosen. In addition, it has a strong positive correlation with WBGT Liljegren and low MAE. In addition, the T g method for WBGT Brimicombe can be suggested to be equivalent to that of WBGT Liljegren enhancing understanding of the relationship of different forms of radiation. Further, WBGT Brimicombe has a strong linear relationship between it's observed and reanalysis data and at most an MAE of 5°C.
WBGT ACSM87 is not an accurate approximation of WBGT and should not be continued to be used. WBGT ACSM87 often has a three heat stress category difference to WBGT Liljegren and it widely has anomalies of ±4°C for the three heatwave case studies chosen. Although WBGT ACSM87 has a strong positive correlation with WBGT Liljegren , it shows high MAE values.
It is hoped that by integrating WBGT Brimicombe into reanalysis, climate models and forecasts, that this information would be made openly accessibly and incorporated into sectors heat warning and adaptations, providing improvements to early warning systems and adaptation policy. Finally, WBGT Brimicombe is a worthy heat stress index candidate for a global forecasting early warning system and would not only be beneficial to a range of sectors but also has the real potential to save lives.